The Relationship Between Augmented Lyapunov-Krasovskii Functionals and Estimated Inequalities

Different estimated inequalities and augmented Lyapunov-Krasovskii functionals (LKFs) play important role in assessing the stability of time-delay systems. In this technical note, three categories of estimated inequalities are introduced, in which either all matrices, some matrices or no matrices are free. Then, the internal relationship among the three categories of estimated inequalities is fully revealed. Next, an optimal method is provided for selecting the estimated inequalities and constructing the Lyapunov-Krasovskii functionals. That is, the inequalities and the functionals should tailor for each other (see Table 2), which is proved theoretically. Finally, a numerical example is presented to verify the results.